The on-shell action of supergravity & the B-side of TsT and single-trace TT

A bstract We propose boundary terms for the action of the NS sector of supergravity on M 3 × S 3 × T 4 spacetimes — where M 3 is an AdS 3 or a linear dilaton background — that render the Brown-York stress tensor and the on-shell action finite. For AdS 3 backgrounds, we show that the on-shell action yields a free energy with chemical potentials determined by the B -field. TsT (T-duality + shift + T-duality) transformations of these backgrounds generate classes of linear dilaton backgrounds distinguished by their boundary conditions. Among them, there is one class of backgrounds that reproduces features of the single-trace TT T T ¯ deformation. These backgrounds have additional chemical potentials that can be turned off by large gauge transformations of the B -field. We show that the Brown-York stress tensor of these backgrounds reproduces the energies and the trace flow equation of TT T T ¯ -deformed CFTs. We also show that the on-shell action matches the TT T T ¯ partition function and discuss the interpretation of these results.